I am working on an optimization problem, which is an MINLP (with binary integers). Is this MINLP an NP-Hard problem or NP-Complete problem. And how to prove the hardness or completeness? Here $c_{k,n}$ and $p_{k,n}$ are optimization variables.
Here $\mathcal{N}=\{1,2,\cdots,N\}$ and
$\mathcal{K}=\{1,2,\cdots,K\}$
$\underset{\{{{c}_{k,n}},{{p}_{k,n}}\}}{\mathop{\max }}\,\sum\limits_{k=1}^{K_1}{\sum\limits_{n=1}^{N}{{{c}_{k,n}}\{\frac{1}{N}{{\rm{log_2}}\left( 1+p_{k,n}h_{k,n} \right)}}}\}$
subject to
$\text{}\hspace{1mm}\text{C1: }{{c}_{k,n}}\in \left\{ 0,1 \right\},\hspace{8mm}\forall k,n,\hspace{1mm}k\in\mathcal{K},\hspace{1mm}n\in\mathcal{N}$
$\text{}\hspace{1mm} \text{C2: }\sum\limits_{k=1}^{{K}}c_{k,n}=1,\hspace{9mm} \forall n,,\hspace{1mm}n\in\mathcal{N}$
$\text{}\hspace{1mm} \text{C3: }{{p}_{k,n}}\ge 0,\hspace{13.5mm}\forall k,n,\hspace{1mm}k\in\mathcal{K},\hspace{1mm}n\in\mathcal{N} $
$\text{}\hspace{1mm} \text{C4: }\sum\limits_{k=1}^{{K}}{\sum\limits_{n=1}^{N}{{{c}_{k,n}}{{p}_{k,n}}\le {P_{\rm{Max}}}}} $
$\text{}\hspace{1mm} \text{C5: }\frac{1}{N}\sum\limits_{n=1}^{N}{c}_{k,n}{\rm{log_2}}\left( 1+p_{k,n}h_{k,n} \right)\ge R_k, \hspace{1mm}k=K_1+1,\cdots, {K} $